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Long Time Behavior of a Point Mass in a One-Dimensional Viscous Compressible Fluid and Pointwise Estimates of Solutions. (arXiv:1904.00992v2 [math.AP] UPDATED)

We consider the motion of a point mass in a one-dimensional viscous
compressible barotropic fluid. The fluid--point mass system is governed by the
barotropic compressible Navier--Stokes equations and Newton's equation of
motion. Our main result concerns the long time behavior of the fluid and the
point mass; it gives pointwise convergence estimates of the density and the
velocity of the fluid to their equilibrium values. As a corollary, it shows
that the fluid velocity $U(x,t)$ and the point mass velocity $V(t)=U(h(t)\pm
0,t)$, where $h(t)$ is the location of the point mass, decay differently as
$||U(\cdot,t)||_{L^{\infty}(\mathbb{R}\backslash \{ h(t) \})}\approx t^{-1/2}$
and $|V(t)|\lesssim t^{-3/2}$. This discrepancy between the decay rates of
$||U(\cdot,t)||_{L^{\infty}(\mathbb{R}\backslash \{ h(t) \})}$ and
$|V(t)|=|U(h(t)\pm 0,t)|$ is due to the hyperbolic-parabolic nature of the
problem: The fluid velocity decays slower on the characteristics $x=\pm ct$,
where $c$ is the speed o查看全文